<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:11"/>
( <font color="Maroon">T</font> is <a href="compts_1.html#V2" target="_self">compact</a> implies for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">F</font> is <a href="tops_2.html#V2">closed</a> &amp;  <a href="setfam_1.html#K6">meet</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> ex <font color="Olive">G</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="setfam_1.html#K6">meet</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">T</font> is <a href="compts_1.html#V2" target="_self">compact</a>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:11_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:11_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 <b>and </b><br/><a NAME="E4:11_1"/><i><font color="Green">E35</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E5:11_1"/>
not <font color="Maroon">F</font> is <a href="compts_1.html#V1" target="_self">centered</a>
 
<b>by </b><i>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:11_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:11_1"/>
<font color="Maroon">G</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b><font color="Maroon">X</font> = <font color="Maroon">G</font>;<br/>

<b>thus </b><a NAME="E10:11_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">F</font> is <a href="tops_2.html#V2">closed</a> &amp;  <a href="setfam_1.html#K6">meet</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> ex <font color="Olive">G</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="setfam_1.html#K6">meet</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E3:11"/>
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">F</font> is <a href="compts_1.html#V1" target="_self">centered</a> &amp; <font color="Olive">F</font> is <a href="tops_2.html#V2">closed</a> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:11_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> is <a href="compts_1.html#V1" target="_self">centered</a>
 <b>and </b><br/><a NAME="E2:11_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 ;<br/>

<b>assume </b><a NAME="E3:11_2"/><i><font color="Green">E40</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E4:11_2"/>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:11_2"/><b>then </b>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="setfam_1.html#K6">meet</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:11_2"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
<font color="Maroon">T</font> is <a href="compts_1.html#V2" target="_self">compact</a>
 <b>by </b><i/>;<br/>


</div>
